On Waring's Problem: Two Cubes and Two Minicubes

TL;DR

This paper investigates a variant of Waring's problem, showing that almost all positive integers can be expressed as the sum of four cubes with two small cubes, and provides an asymptotic count for such representations within a specific range of parameters.

Contribution

It proves that almost every positive integer can be represented as the sum of four cubes with two small cubes, extending Waring's problem with new bounds and asymptotic formulas.

Findings

Almost all positive integers are sums of four cubes with two at most $n^{ heta}$ for $ heta \\geq 192/869.

An asymptotic formula is derived for the number of such representations when $1/4<\theta<1/3$.

The results extend classical Waring's problem by incorporating size constraints on some of the cubes.

Abstract

We establish that almost every positive integer $n$ is the sum of four cubes, two of which are at most $n^{\theta}$, as long as $\theta\geq192/869$. An asymptotic formula for the number of such representations is established when $1/4<\theta<1/3$.